On the Amplitude-Frequency Characteristic of a Resonant Amplifier

E. I. Fialko

(Presented by Doctor of Technical Sciences L. S. Gudkin)

Statement of the Problem

As is known [1], the resonance characteristic of an ordinary oscillatory circuit is described by the equation

\[ y=\frac{1}{\sqrt{1+x^2}}; \tag{1} \]

here \(x\) is the generalized detuning, equal to

\[ x=\frac{\Delta f}{d\cdot f_0}\left(1+\frac{f_0}{f}\right), \tag{2} \]

where

\[ \begin{aligned} f_0 &\text{— the resonant frequency of the circuit,}\\ f &\text{— the current frequency,}\\ \Delta f=f-f_0 &\text{— the absolute detuning of the circuit,}\\ d &\text{— the attenuation of the circuit.} \end{aligned} \]

The curve \(y=\varphi(f)\) is asymmetric with respect to the perpendicular erected to the frequency axis at the point corresponding to the resonant frequency \(f_0\) (Fig. 1). The pass band is, naturally, also asymmetric with respect to \(f_0\): the left part of the pass band is smaller than the right. This asymmetry is the more substantial, the greater the ratio of the pass band to the resonant frequency.

In the case of narrow-band systems \((\Delta F \ll f_0)\), within detunings \(\Delta f\) comparable with the pass band \(\Delta F\), the asymmetry of the resonance curve appears very weakly.

Indeed, if \(\Delta f \ll f_0\), then

\[ \frac{f_0}{f}\simeq 1 \]

and the generalized detuning is equal to

\[ x\simeq \frac{2\Delta f}{d\cdot f_0}. \tag{3} \]

The curve \(y=f(f)\) becomes symmetric with respect to \(f=f_0\) (i.e., \(y=f(f)\) is symmetric for small detunings in comparison with \(f_0\), but comparable with the pass band).

It is precisely for this reason that the resonance characteristic is very often considered as a symmetric curve, and such a consideration gives sufficiently accurate results.

For example, it is known that in this case the pass band of the circuit at the level 0.7 is equal to

\[ \Delta F=d\cdot f_0, \tag{4} \]

and the bandwidth of an \(n\)-stage resonant amplifier built from identical circuits is determined by the formula

\[ \Delta F_n=\Delta F\cdot \sqrt{\sqrt[n]{2}-1}. \tag{5} \]

Fig. 1. Resonance characteristic of an oscillatory circuit.

Naturally, the question arises as to how the asymmetry of the circuit’s resonance curve affects the properties of a multistage amplifier; whether formulas (4) and (5) remain valid in this case as well; and whether, in the case of wideband systems, the asymmetry of the circuit’s resonance curve may be neglected.

To answer these and also certain other questions, let us determine:

1. How is the bandwidth expressed in terms of the damping of the circuit (for one stage and for a multistage amplifier)?
2. What is the degree of asymmetry of the passband?
3. What is the limiting form of the amplitude-frequency characteristic of a multistage amplifier for a very large number of stages?

In the present work these questions are considered for the case of a multistage resonant amplifier built from identical circuits.

Let us first consider the amplitude-frequency characteristic of a single stage, or the resonance curve of an oscillatory circuit.

Oscillatory Circuit

As is known [1], the resonance characteristic of an oscillatory circuit is described by equation (1). Let us define the bandwidth of the circuit at the level \(y \simeq 0.7\). To do this, we determine the values of the positive and negative detunings \((\Delta f_{0.7(+)}\) and \(\Delta f_{0.7(-)})\) corresponding to the level \(0.7\) (Fig. 2).

Putting

\[ y=\frac{1}{\sqrt{2}}\simeq 0.7, \]

we obtain

\[ x_{0.7}=\pm 1, \]

i.e.,

\[ \frac{\Delta f_{0.7}}{d\cdot f_0} \left( 1+\frac{f_0}{f_0+\Delta f_{0.7}} \right) =\pm 1 \]

or

\[ \Delta f_{0{,}7}^{2}+\Delta f_{0{,}7}\cdot f_{0}(2\pm d)\pm d f_{0}^{2}=0. \tag{6} \]

Solving equation (6) with respect to \(\Delta f_{0{,}7}\), we obtain:

\[ \Delta f_{0{,}7} = -f_{0}\left(1\pm\frac{d}{2}\right) \pm \sqrt{ f_{0}^{2}\left(1\pm\frac{d}{2}\right)^{2} \mp d f_{0}^{2} }. \]

Fig. 2. Resonance characteristic of an oscillatory circuit in the region of detunings comparable with the passband.

It is not difficult to see that in the last expression only the plus sign must be taken before the radical.

Indeed, when \(x_{0{,}7}=+1\), the detuning \(\Delta f_{0{,}7}\) must also be positive (2), and this condition is satisfied only in the case when the solution with the plus sign before the radical is taken.

Thus, the positive and negative detunings are respectively equal to:

\[ \left. \begin{aligned} \Delta f_{0{,}7(+)} &= -f_{0}\left(1-\frac{d}{2}\right) + \sqrt{ f_{0}^{2}\left(1-\frac{d}{2}\right)^{2} +d f_{0}^{2} } \\ \Delta f_{0{,}7(-)} &= -f_{0}\left(1+\frac{d}{2}\right) + \sqrt{ f_{0}^{2}\left(1+\frac{d}{2}\right)^{2} -d f_{0}^{2} } \end{aligned} \right\}. \]

After simple transformations we obtain:

\[ \begin{aligned} \Delta f_{0.7(+)} &= -f_0\left(1-\frac{d}{2}\right)+f_0\sqrt{1+\frac{d^2}{4}},\\ \Delta f_{0.7(-)} &= -f_0\left(1+\frac{d}{2}\right)+f_0\sqrt{1+\frac{d^2}{4}}. \end{aligned} \tag{7} \]

The passband at the level 0.7 is equal to

\[ \Delta F_{0.7}=\Delta f_{0.7(+)}-\Delta f_{0.7(-)}=d\cdot f_0 . \tag{8} \]

As we see, the passband of the real resonance curve, which possesses a definite asymmetry, is equal to the product of the damping and the resonant frequency. As is known [1], in the case of a strictly symmetric resonance curve described by formulas (1) and (3), the passband is also equal to the product \(d\cdot f_0\).

The degree of asymmetry of the passband may be estimated by the ratio

\[ \psi=\frac{\Delta f_{0.7(+)}+\Delta f_{0.7(-)}}{\Delta F_{0.7}} . \tag{9} \]

Substituting into (9) the corresponding expressions from (7) and (8) and canceling numerator and denominator by \(f_0\), we obtain:

\[ \psi=2\,\frac{\sqrt{1+\dfrac{d^2}{4}}-1}{d}. \]

For the case \(\dfrac{d^2}{4}\ll 1\) we have:

\[ \psi \simeq \frac{d}{4} \tag{10} \]

or

\[ \psi \simeq \frac{\Delta F_{0.7}}{4 f_0}. \]

Thus, the degree of asymmetry of the passband is uniquely determined by the damping of the circuit. It was assumed above that the loss resistance of the circuit does not change under detuning and remains equal to the resistance at the resonant frequency \(f_0\). In reality, the components of the active resistance of the circuit depend on the magnitude of the detuning \(\Delta f\).

In the general case, the active losses of the circuit include the following components:

1) the intrinsic active losses of the circuit;
2) losses introduced by the input resistance of the tube of the following stage (in a multistage amplifier);
3) losses introduced by an additional resistance (as is known, in wideband circuits the oscillatory circuit is shunted by an active resistance).

The shape of the resonance curve of the circuit will depend on the character of the variation of each of these components (as a function of frequency).

An analysis of this dependence is not part of the task of the present work; therefore, confining ourselves here to simple physical reasoning and taking into account the dependence of the magnitude of the intrinsic losses of the circuit, the input resistance, etc., on frequency, we conclude that the right-hand branch of the resonance curve will descend, and the left-hand branch will rise, in comparison with the case when the loss resistance is constant. One should also bear in mind the influence of feedba—

...couplings on the shape of the resonance curve of the cascade. In order to determine how sound the investigation being carried out is, which assumes the constancy of the active-loss resistance, it is necessary to turn to experiment. Experiment has shown that the resonance characteristic of the cascade retains an asymmetric character (see below).

Let us proceed to the consideration of a multistage resonant amplifier.

Multistage Resonant Amplifier

The equation of the amplitude-frequency characteristic of a multistage resonant amplifier built on circuits with identical damping and resonant frequencies¹, as is known [1], has the form

\[ y_n=\left(\frac{1}{\sqrt{1+x^2}}\right)^n, \]

where

\(n\) is the number of stages, and \(x\) is the generalized detuning. Let us determine the passband of such an amplifier at the level \(y_n \simeq 0.7\). From the equation

\[ \left(\frac{1}{\sqrt{1+x^2}}\right)^n=\frac{1}{\sqrt{2}} \]

we find the generalized detuning corresponding to the level 0.7:

\[ x_{0.7}=\pm \sqrt{\sqrt[n]{2}-1}. \tag{11} \]

Introducing the notation

\[ a=\pm \sqrt{\sqrt[n]{2}-1} \]

and substituting into (11), in place of \(x_{0.7}\), its expanded expression, we obtain

\[ \frac{\Delta f_{0.7}}{d f_0} \left(1+\frac{f_0}{f_0+\Delta f_{0.7}}\right)=a. \tag{12} \]

The latter expression is a quadratic equation with respect to the detuning \(\Delta f_{0.7}\) and, after simple transformations, is brought to the form:

\[ \Delta f_{0.7}^{\,2}+\Delta f_{0.7} f_0(2-a d)-a d f_0^{\,2}=0. \tag{13} \]

Solving equation (13), we obtain:

\[ \Delta f_{0.7}=f_0\left(\frac{a d}{2}-1\right) \pm \sqrt{ f_0^{\,2}\left(\frac{a d}{2}-1\right)^2 +a f_0^{\,2} d }. \]

It is easy to see that the second solution, with the minus sign before the radical (in the last formula), is devoid of physical meaning. Indeed, if the reduced detuning is positive \((x_{0.7}=+|a|)\), then, as follows from (12), the absolute detuning must also be positive (note that always \(\Delta f_{0.7}>-f_0\)). Therefore only the first solution is acceptable, namely:

\[ \Delta f_{0.7}=f_0\left(\frac{a d}{2}-1\right) + \sqrt{ f_0^{\,2}\left(\frac{a d}{2}-1\right)^2 +a f_0^{\,2} d }= \]

\[ = f_0\left(\frac{a.d}{2}-1\right)+f_0\sqrt{1+\left(\frac{a.d}{2}\right)^2}. \tag{14} \]

But the coefficient \(\alpha\) has 2 values:

\[ \alpha_1=|\alpha|, \]
\[ \alpha_2=-|\alpha|. \]

Substituting \(\alpha_1\) and \(\alpha_2\) into (14), we find the positive and negative detunings:

\[ \begin{aligned} \Delta f_{0.7(+)}&=f_0\left(\frac{|\alpha|.d}{2}-1\right)+f_0\sqrt{1+\left(\frac{\alpha.d}{2}\right)^2},\\ \Delta f_{0.7(-)}&=f_0\left(-\frac{|\alpha|.d}{2}-1\right)+f_0\sqrt{1+\left(\frac{\alpha.d}{2}\right)^2}. \end{aligned} \]

The passband is equal to

\[ \Delta F_{0.7}=\Delta f_{0.7(+)}-\Delta f_{0.7(-)}=|\alpha|.d.f_0, \tag{15} \]

i.e.

\[ \Delta F_{0.7}=d.f_0.\sqrt{\sqrt[n]{2}-1} \tag{16} \]

and, for a sufficiently large number of stages,

\[ \Delta F_{0.7}\simeq d.f_0.\sqrt{\frac{\ln 2}{n}}. \]

It is known [1] that, in the case of an \(n\)-stage resonant amplifier whose circuits are characterized by strictly symmetrical curves, the passband is also determined by formula (16). The asymmetry of the passband is estimated by the coefficient

\[ \psi=\frac{\Delta f_{0.7(+)}+\Delta f_{0.7(-)}}{\Delta F_{0.7}} =2\frac{\sqrt{1+\left(\frac{\alpha.d}{2}\right)^2}-1}{|\alpha|.d}. \tag{17} \]

For

\[ \left(\frac{\alpha.d}{2}\right)^2\ll 1 \]

\[ \psi\simeq \frac{|\alpha|.d}{4}, \tag{18} \]

and in the case \(n\gg 1\)

\[ \psi\simeq \frac{d}{4}\sqrt{\frac{\ln 2}{n}}. \tag{19} \]

Comparing formulas (19) and (10), we conclude that the degree of asymmetry of the passband of the entire amplifier decreases as the number of stages increases. In other words, the amplitude-frequency characteristic of a multistage resonant amplifier is less asymmetric than the resonance curves of the stages of which the amplifier consists. In the case, however, when the passband of the multistage amplifier is kept constant as the number of stages changes, the degree of asymmetry of the passband does not change in the first approximation. Indeed, expressing from (15) the attenuation of a stage through the resultant passband, we have

\[ d=\frac{\Delta F_{0.7}}{f_0.|\alpha|}. \]

Substituting the last expression into (18), we obtain

\[ \psi \simeq \frac{\Delta F_{0.7}}{4 f_0}. \tag{20} \]

Let us now find the limiting form of the amplitude–frequency characteristic of the amplifier as the number of stages is increased while the resulting passband is kept unchanged.

For this purpose, in the equation for the amplitude–frequency characteristic of the amplifier

\[ y_n=\left(\frac{1}{\sqrt{1+\left[\frac{\Delta f}{d f_0}\left(1+\frac{f_0}{f_0+\Delta f}\right)\right]^2}}\right)^n \tag{21} \]

we express \(d f_0\) in terms of the resulting passband \(\Delta F_n\) and the number of stages.

Remark. Here and below the index “0.7” is omitted in the expression for the passband.

From (16) we have

\[ d f_0=\frac{\Delta F_n}{\sqrt{\sqrt[n]{2}-1}}. \tag{22} \]

Substituting (22) into (21), we obtain

\[ y_n=\left\{1+\left[\frac{\Delta f}{\Delta F_n}\cdot \sqrt{\sqrt[n]{2}-1}\cdot \left(1+\frac{f_0}{f_0+\Delta f}\right)\right]^2\right\}^{-\frac{n}{2}}. \]

For sufficiently large \(n\),

\[ y_n\simeq \left\{1+\left[\frac{\Delta f}{\Delta F_n} \left(1+\frac{f_0}{f_0+\Delta f}\right)\right]^2 \frac{\ln 2}{n}\right\}^{-\frac{n}{2}}. \tag{23} \]

We find the limiting form of the amplitude–frequency characteristic by passing to the limit

\[ y=\lim_{n\to\infty} y_n. \tag{24} \]

Introducing

\[ \varepsilon= \left[\frac{\Delta f}{\Delta F_n} \left(1+\frac{f_0}{f_0+\Delta f}\right)\right]^2 \cdot \frac{\ln 2}{n} \]

and taking into account that as \(n\to\infty\), \(\varepsilon\to 0\), from (23) and (24) we obtain

\[ y=\lim_{\varepsilon\to 0}(1+\varepsilon)^{\frac{1}{\varepsilon}\cdot \left\{-\frac{\ln 2}{2}\left[ \frac{\Delta f}{\Delta F_n} \left(1+\frac{f_0}{f_0+\Delta f}\right)\right]^2\right\}} = \]

\[ =e^{-\frac{\ln 2}{2}\left[ \frac{\Delta f}{\Delta F_n} \left(1+\frac{f_0}{f_0+\Delta f}\right)\right]^2}. \tag{25} \]

In the case of small detunings (or, what is the same, in the case where the characteristic of the circuit is assumed to be a symmetric curve), i.e., for

\[ \frac{f_0}{f_0+\Delta f}=1, \]

the amplitude–frequency characteristic takes the form

\[ y^*=e^{-\frac{\ln 2}{2}\left(\frac{2\Delta f}{\Delta F_n}\right)^2}. \tag{26} \]

Let us note that the limiting equation of the amplitude-frequency characteristic of a multistage resonant amplifier was obtained by V. I. Siforov in the form [2]

\[ y=e^{-z^{2}} . \]

Expression (26) may be regarded as an approximate limiting equation of the real, asymmetric amplitude-frequency characteristic, whereas equation (25) is exact\(^1\).

Between the approximate and the exact representations of the limiting form of the amplitude-frequency characteristic of a resonant amplifier, as is easy to see, there is the relation

\[ y=(y^{*})^{\xi^{2}}, \]

where

\[ \xi=\frac{1}{2}\left(1+\frac{f_{0}}{f_{0}+\Delta f}\right) \]

or

\[ \xi=\frac{1+\dfrac{\Delta f}{2 f_{0}}}{1+\dfrac{\Delta f}{f_{0}}}. \]

Let us note that for

\[ \begin{aligned} \Delta f &=0 && \xi=1;\\ \Delta f &>0 && \xi<1;\\ \Delta f &<0 && \xi>1. \end{aligned} \]

In conclusion, let us note that passage to the limit for an infinitely large number of stages is purely formal. However, the limiting form of the amplitude-frequency characteristic reflects the properties of the amplifier quite well even for a comparatively small number of stages, for example, for \(n>3\) [3]. As an illustration, let us give the characteristic of a five-stage resonant amplifier with bandwidth \(\Delta F=3\) mc/s, tuned to the frequency \(f_{0}=30\) mc/s,

\[ y_{5}=\frac{1}{\left(\sqrt{1+x^{2}}\right)^{5}}, \]

where, as follows from (22) and (2),

\[ x=\frac{\Delta f}{\Delta F}\cdot \sqrt{\,\sqrt[5]{2}-1\,}\cdot \left(1+\frac{f_{0}}{f}\right). \]

This characteristic is shown in Fig. 3 (curve \(y_{5}\)). The same graph also shows the limiting amplitude-frequency characteristic constructed from formula (25) (curve \(y\infty\)).

The greatest deviation of \(y_{5}\) from \(y\infty\) within the pass band does not exceed 2%.

Experimental Verification of Some Results

The investigation presented above was carried out under the assumption that the resistance of the active losses of the circuit is independent of the magnitude of the detuning. Moreover, the influence of feedbacks, which—

\(^1\) With the resistance of the losses constant.

... can substantially change the shape of the amplitude-frequency characteristic both of a single stage and of the entire amplifier. Therefore one should not expect exact agreement between the experimental data and the results obtained above.

Fig. 3. Amplitude-frequency characteristic of a five-stage resonant amplifier \((y_5)\) and the limiting amplitude-frequency characteristic of a multistage resonant amplifier \((y_\infty)\).

In order to evaluate the conclusions relating to multistage amplifiers, it is first of all necessary to determine how valid the initial assumption \((r_{\text{loss}} = \text{constant})\) is in the case of a single stage.

For this purpose, the amplitude-frequency characteristic of a broadband stage of a resonant amplifier was measured.

The circuit of the stage is shown in Fig. 4, and the amplitude-frequency characteristic of the stage in Fig. 5.

The bandwidth of the stage is \(\Delta F_{0.7} = 10.7\) MHz, and the resonant frequency is \(f_0 = 30\) MHz. As is seen from the characteristic, the right and left parts of the bandwidth are respectively equal to:

\[ \Delta f_{0.7(+)} = 5.7\ \text{MHz}, \]

\[ \Delta f_{0.7(-)} = -5\ \text{MHz}. \]

Thus, the asymmetry of the passband is estimated by the coefficient

\[ \psi = \frac{\Delta f_{0.7(+)} + \Delta f_{0.7(-)}}{\Delta F_{0.7}} = \frac{0.7}{10.7} \simeq 6.5\%. \]

Fig. 4. Stage of a broadband resonant amplifier.

As was shown above, the coefficient \(\psi\) is equal to

\[ \psi \simeq \frac{d}{4} \qquad \left(\text{provided that } \frac{d^2}{4} \ll 1\right), \]

where

\[ d=\frac{\Delta F_{0.7}}{f_0}. \]

Fig. 5. Amplitude-frequency characteristic of a stage of a broadband resonant amplifier.

In our case

\[ d=\frac{10.7}{30}=0.356 \qquad \left(\frac{d^{2}}{4}\simeq 0.03 \ll 1\right). \]

Thus,

\[ \psi \simeq \frac{d}{4}=8.9\%. \]

Thus, the discrepancy between the result of the experiment and of the analysis is \(\simeq 30\%\), and the asymmetry of the experimentally recorded amplitude-frequency characteristic proved to be less sharply expressed than followed from the analysis.

This result is quite understandable; as was indicated, the increase in the resistance of active losses with increasing frequency leads to the fact that the right-hand branch of the characteristic descends, while the left-hand branch rises.

Comparing the data of analysis and experiment, we arrive at the following conclusion: first, the asymmetry of the amplitude-frequency characteristic can be estimated by formulas 17, 18, 19, bearing in mind that these formulas give a somewhat overestimated value of the asymmetry coefficient. Second, the asymmetry of the amplitude-frequency characteristic within the limits of the passband is small.

Conclusions

1. The asymmetry of the resonance characteristic of a circuit naturally leads to asymmetry of the amplitude-frequency characteristic of a multistage amplifier, and also to asymmetry of the passband. However, the magnitude of the passband will be the same as in the case of a symmetric resonance curve of a circuit with the given damping. For an \(n\)-stage resonant amplifier the passband at the level 0.7 is equal to

\[ \Delta F_n=d f_0 \cdot \sqrt{\sqrt[n]{2}-1} \]

and for \(n \gg 1\)

\[ \Delta F_n \simeq d f_0 \cdot \sqrt{\frac{\ln 2}{n}}. \]

1. An estimate of the asymmetry of the passband is given by formulas (17)—(20). In particular, for a large number of stages and small damping

\[ \psi \simeq \frac{d}{4}\cdot \sqrt{\frac{\ln 2}{n}}. \]

1. The limiting form of the amplitude-frequency characteristic of a multistage resonant amplifier for \(n \to \infty\) and a specified resultant passband at the level 0.7 \((\Delta F)\) will be:

\[ y=e^{-2\cdot \ln 2 \cdot \left[\frac{\Delta f}{\Delta F}\right]^2 \cdot \left(\frac{1+\frac{\Delta f}{2 f_0}}{1+\frac{\Delta f}{f_0}}\right)^2}. \]

For the case of small detunings or under the assumption of symmetry of the resonance curve of the circuit, as is known,

\[ y=e^{-2\cdot \ln 2 \cdot \left[\frac{\Delta f}{\Delta F}\right]^2}. \]

References

1. V. I. Siforov. Radio Receiving Devices. Voenizdat, 1951.
2. V. I. Siforov. Noise in an Amplifier. ONTI, 1936.
3. R. D. Leites. Transient processes in a multistage resonant amplifier. Radiotekhnika, No. 3, 1947.

1. In what follows, such circuits are called equal or identical. ↩